Thursday, September 18, 2014

Hypermodern Classical Mechanics

Really great paper from Norden Huang of Hilbert-Huang Transform (HHT) fame. The opening sentence is great:

"Historically, there are two views of nonlinear mechanics: the Fourier and the Poincare. ...

This is thus the Fourier view: The system has a fundamental oscillation (the first-order solution) and bounded harmonics (all the higher-order solutions). Although this approach might be mathematically sound, and seems to be logical, the limitations of this view become increasingly clear on closer examination: First, the perturbation approach is limited to only small nonlinearity; when the nonlinear terms become finite, the perturbation approach then fails; Second, and more importantly, the solution obtained makes little physical sense. It is easily seen that the properties of a nonlinear equation should be different from a collection of linear ones; therefore, the two sets of solutions from the original equation and the perturbed ones should have different physical and mathematical properties.
...
Poincare's system provides a discrete description. It defines the mapping of the phase space onto itself. In many cases, Poincare mapping enables a graphical presentation of the dynamics. Typically, the full nonlinear solution is computed numerically. Then the dynamics are viewed through the intersections of the trajectory and a plane cutting through the path in the phase space. The intersections of the path and the plane are examined to reveal the dynamical characteristics. This approach also has limitations, for it relies heavily on the periodicity of the processes. The motion between the Poincare cuts could also be just as important for the dynamics. Both the Fourier and Poincare views have existed for a long time.

Only recently has an alternative view for mechanics, the Hilbert view, been proposed."

Okay, so it isn't surprising that Huang would see the world through an HHT filter, but still it's great to realize what I thought of as being a cool trick become the foundation of a worldview! If you're familiar with the HHT, then you can skip to page 22 (that is, 442 if your going by page labels) where Huang et. al. start going through the motions on real nonlinear systems. It's amusing that in spite of the fact that this is a paper that promises to be about fairly hardcore math, someone felt the need to explain what factoring x meant. The punchline is in the last paragraph of this section: "The Hilbert representation gives a true physical interpretation of the dynamics by indicating the instantaneous value of the frequency, and thus the Hilbert view is much more physical.".

The supposed increased physical insight begins to be used on page 30 (that is, 440), where Stokes waves are examined with HHT instead of Fourier transforms. The wave theory (perhaps, "point of view" would be better) of water waves is found to be insufficient: "In numerous controlled experiment ... the main frequency is seen to shift to a lower frequency ...". Obviously, the wave theory says the frequencies should stay still. They explain: "This decrease, however, occurs only at certain very local regions. Such local change makes the data nonstationary, the condition Fourier analysis is ill-equipped to deal with. ... [T]he Fourier spectral peak represents only the global mean frequency. It is not sensitive to the local change of frequency as in the local and discrete downshift phenomena.". The weakness of Fourier methods is connected to their linearity and the consequent uncertainty principle: "the only way the Fourier method can represent a local frequency change is through harmonics. But such a representation is no longer local.".

I think that Huang here makes a good argument that HHT is a good first tool, because it doesn't require a lot of assumptions and computing power is cheap. Huang definitely shows that there are situations where interpreting the Fourier Spectra physically is difficult, and perhaps they in these situations are purely formal. The local, non-linear nature of the HHT is an advantage when systems aren't linear. So, HHT has no advantages for Quantum Mechanics. However, this article is much too empirical (a rare problem indeed!). Advantages and disadvantages are shown by example, but how can anybody know if these are just special cases. In addition, the Poincare and Fourier approaches aren't antithetical. In order to demonstrate greatness, the interconnection between this Hilbert view needs to be explored. Some fresh uncertainty principle would be nice! Also, the Poincare and Fourier systems views give a misleading picture because of their global nature, but they are motivated by deep physical theory - Fourier Analysis by the Newton's 3 Laws formulation of classical mechanics (everything is second order DEs) and Poincare by the Lagrangian/Hamiltonian view of classical mechanics (everything is motion in phase space). The physical motivation of the non-linear HHT should be more deeply delved into. It's given some room in the first part I said one could skip, but I think it should have been more prominent (perhaps I should read Bendat & Piersol ...). The local nature of HHT would make more sense if this was done, I think.

Anyway, this is a nice little paper full of neat applications of a cool modern scientific tool. A good distraction from both my real jobs and my quantum mechanics blogging (which I promise will happen). Enjoy!